Hello there. Hope you are doing good. I would like to tell you that there is no loss if you do a degree from private university or a government university. There is no restriction on appearing for any exam. The only criteria that is mentioned for every exam is that the degree should be from a recognized university. If your degree is from a recognized university, you won't face any issue in job or appearing for any exam.

I hope you found this answer helpful. Good luck for your future.